Find the mean, variance and standard deviation for the following data: 6,7,10,12,13,4,8,12

Explanation

To find the mean, variance, and standard deviation for the given data: 6, 7, 10, 12, 13, 4, 8, and 12, we will follow these steps:

### Step 1: Calculate the Mean
The mean (average) is calculated using the formula:
\[ \text{Mean} (\bar{x}) = \frac{1}{n} \sum_{i=1}^{n} x_i \]
where \( n \) is the number of observations and \( x_i \) are the data points.

1. **Sum the data points**:
\[ 6 + 7 + 10 + 12 + 13 + 4 + 8 + 12 = 72 \]

2. **Count the number of data points**:
\[ n = 8 \]

3. **Calculate the mean**:
\[ \bar{x} = \frac{72}{8} = 9 \]

### Step 2: Calculate the Variance
Variance is calculated using the formula:
\[ \text{Variance} (\sigma^2) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]

1. **Calculate each \( (x_i - \bar{x})^2 \)**:
- For \( x_1 = 6 \): \( (6 - 9)^2 = (-3)^2 = 9 \)
- For \( x_2 = 7 \): \( (7 - 9)^2 = (-2)^2 = 4 \)
- For \( x_3 = 10 \): \( (10 - 9)^2 = (1)^2 = 1 \)
- For \( x_4 = 12 \): \( (12 - 9)^2 = (3)^2 = 9 \)
- For \( x_5 = 13 \): \( (13 - 9)^2 = (4)^2 = 16 \)
- For \( x_6 = 4 \): \( (4 - 9)^2 = (-5)^2 = 25 \)
- For \( x_7 = 8 \): \( (8 - 9)^2 = (-1)^2 = 1 \)
- For \( x_8 = 12 \): \( (12 - 9)^2 = (3)^2 = 9 \)

2. **Sum these squared differences**:
\[ 9 + 4 + 1 + 9 + 16 + 25 + 1 + 9 = 74 \]

3. **Calculate the variance**:
\[ \sigma^2 = \frac{74}{8} = 9.25 \]

### Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[ \text{Standard Deviation} (\sigma) = \sqrt{\sigma^2} = \sqrt{9.25} \approx 3.04 \]

### Final Results
- Mean: \( 9 \)
- Variance: \( 9.25 \)
- Standard Deviation: \( 3.04 \)